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On the dynamics of the transition to vortex breakdown in axisymmetric inviscid swirling flows
Accepted Manuscript On the dynamics of the transition to vortex breakdown in axisymmetric inviscid swirling flows M. Vanierschot PII: DOI: Reference:
S0997-7546(16)30101-7 http://dx.doi.org/10.1016/j.euromechflu.2017.02.009 EJMFLU 3144
To appear in:
European Journal of Mechanics B/Fluids
Received date: 22 March 2016 Please cite this article as: M. Vanierschot, On the dynamics of the transition to vortex breakdown in axisymmetric inviscid swirling flows, European Journal of Mechanics B/Fluids (2017), http://dx.doi.org/10.1016/j.euromechflu.2017.02.009 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
2
On the dynamics of the transition to vortex breakdown in axisymmetric inviscid swirling flows
3
M. Vanierschota
1
4 5
6
a
KU Leuven, campus Group T, Andreas Vesaliusstraat 13, B-3000 Leuven, Belgium. Email: [email protected]
Abstract This paper reports on novel features found in the dynamics of the transition to vortex breakdown in inviscid axisymmetric flows with swirl. These features are revealed by a transient simulation of an open ended pipe flow where the inlet swirl is suddenly increased from a swirl number just below the onset of vortex breakdown to a swirl number just above the onset of vortex breakdown. To eliminate the numerous parameters influencing breakdown, the axisymmetric Euler equations with swirl are used as a fluid flow model and solutions are obtained by means of numerical simulation. It is shown that as the step response has died out, the flow evolves to a quasi-static state where time derivatives of variables are negligible small. Stability analysis of this state shows that it can support standing waves in a small region of the flow domain. These standing waves are observed in the simulations as an imbalance in the axial momentum equation which slows down the flow near the central axis. The amplitude of this imbalance grows exponentially in time with a dimensionless growth rate of 0.83 scaled with the flowthrough time. Eventually, the axial velocity along the central axis becomes negative in a small part of the flow, leading to an axisymmetric recirculation zone,
Preprint submitted to International Journal of Heat and Fluid Flow
November 11, 2016
called vortex breakdown. To the authors knowledge, this study would be the first to reveal these features prior to breakdown and the results may help in understanding of the physical mechanisms leading to it as this is still a controversial issue in literature. 7
Keywords: Vortex breakdown, Stability analysis, Inviscid swirling flow
8
1. Introduction
9
Vortex breakdown has fascinated the scientific community for almost six
10
decades now. It has firstly been discovered by Peckham and Atkinson (1957)
11
as the bursting of a leading-edge vortex in the flow over a delta wing. Later it
12
has also been found in numerous other engineering flow cases, amongst others
13
rotating pipe flow, jet flow and enclosed cylinder flow. Despite research for
14
over six decades now, there is still no general explanation for vortex break-
15
down as it is a complex function of numerous flow parameters. Two main
16
general theories exist: the instability theory of Ludwieg (1960) and the tran-
17
sition from a supercritical to a subcritical flow by Benjamin (1962). Ludwieg
18
stated that vortex breakdown is a consequence of hydrodynamic instabilities
19
in the flow which grow in time and eventually lead to breakdown (Ludwieg,
20
1960). Benjamin stated that breakdown is the transition of a supercritical
21
flow (unable to support standing waves) to a subcritical flow (able to support
22
standing waves), similar to the hydraulic jump (Benjamin, 1962). As both
23
theories are unable to predict all the features of breakdown found in experi-
24
ments and simulations, both have found no general acceptance in literature
25
as the main mechanism leading to breakdown.
26
The main reason for the lack of a general theory is the fact that vortex
2
27
breakdown manifests itself in many forms which are depend upon many pa-
28
rameters. In experimental work, no less than 7 types have been identified.
29
The two most commonly observed, called bubble and spiral breakdown, were
30
first reported by Lambourne and Bryer (1961). Bubble or spiral breakdown
31
have been observed in different experiments with similar settings and even
32
transitions between them within the same experiment (without changing the
33
inflow parameters) have been reported. This has led to disagreement on the
34
origin of breakdown (see for instance the review papers of Escudier (1988)
35
and Lucca-Negro and O’Doherty (2001)). Recent studies showed some more
36
insight in the mechanism leading to breakdown. It was found that flows
37
going from below to above the critical state become unstable (Gallaire and
38
Chomaz, 2004; Wang and Rusak, 1996, 1997). Moreover, it has been shown
39
by numerous authors that spiral breakdown occurs in the wake of an ax-
40
isymmetric breakdown as a global instability mode of the flow (Liang and
41
Maxworthy, 2005; Ruith et al., 2003; Gallaire et al., 2006; Meliga et al., 2012;
42
Qadri et al., 2013; Luginsland et al., 2016; Oberleithner et al., 2011). Two
43
modes of spiral breakdown have been observed: the single helix (|m|=1) and
44
the double helix (|m|=2), where m is the azimuthal wave number. A re-
45
cent study of Meliga et al. (2012) showed that both single or double helix
46
breakdown are a bifurcation from axisymmetric breakdown and that mode
47
selection depends on the swirl number.
48
In helping to understand the physical phenomena leading to breakdown, this
49
paper studies the dynamics of an axisymmetric inviscid flow just before the
50
onset of vortex breakdown. Analysis of the axial and radial momentum bal-
51
ances and a stability analysis reveals the mechanisms leading to the forma-
3
52
tion of an axisymmetric breakdown bubble. As such, the results of this study
53
may contribute to more understanding of the physical mechanisms leading
54
to vortex breakdown.
55
2. Governing equations and boundary conditions
56
2.1. Pipe geometry
57
The geometry used in this study is based on the numerical study of Dar-
58
mofal and Murman (1994) and the experimental work of Sarpkaya (1974)
59
and is also similar to the one used in Beran and Culick (1992). A schematic
60
view is shown in figure 1. The geometry is rotational symmetric in the θ di-
61
rection in a cylindrical (r,θ,z)-coordinate system, where r is the radial, θ the
62
azimuthal and z the axial direction. The pipe is divided into four different
63
sections and the radius of the pipe R(z) as a function of the axial direction
64
z is given as
R(z) =
Ri + (Rt − Ri )g(z, zt ), Rt + (Ro − Rt )g(z − zt , zo − zt ),
(0 < z < zt ) (zt < z < zo )
Ro , (zo < z < zc ) R + (R c max − Ro )g(z − zc , zmax − zc ), (zc < z < zmax )
(1)
65
where the function g(z, L) is defined as
66
1 g(z, L) = {1 − cos[π(z/L)]} (2) 2 and the values of the dimensions are given in table 1. The first section
67
(0 < z < zt ) is convergent which ensures a supercritical flow at zt to pre-
68
vent disturbances from traveling upstream towards the inlet section, i.e. it 4
12
10
8
r
S2
6
4
Section II
Section I
S1
Section IV
Section III
S3
2
0
zo
zt
zc
S4
zmax
z
−2
−4
−6
Figure 1: Schematic view of the pipe geometry and the different sections. The radial direction is enlarged by a factor 3 for clarity. The radii in Table 1 corresponding to the axial locations z in the figure have the same index. 0
5
10
15
20
25
30
Table 1: Values of the pipe geometry constants
zi
zt
zo
zc
zmax
0
5
10
25
30
Ri
Rt
Ro
Rc
Rmax
2.0 1.8
2.0
2.0
1.8
69
separates the inlet from the vortex breakdown region. The second section
70
(zt < z < zo ) is diverging to create an adverse pressure gradient on the flow.
71
Adverse pressure gradients are known to promote vortex breakdown (Sarp-
72
kaya, 1974) and hence the location of the bubble lies within or just after this
73
section. The third section (zo < z < zc ) has a constant radius and the last
74
section (zc < z < zmax ) is convergent. This promotes a transition of the
75
flow towards the supercritical state again and hence preventing the output
76
boundary conditions from disturbing the bubble formation.
5
35
40
77
2.2. Governing equations
78
Despite the fact that most swirling flows are highly three-dimensional,
79
breakdown often starts axisymmetric, especially for low Reynolds number
80
flows. After the onset of axisymmetric breakdown, helical instabilities start
81
to develop, creating a three-dimensional flow field (e.g. Ruith et al. (2003)).
82
As this study focuses on the features before breakdown, the axisymmetric
83
variant of the equations are taken were derivatives in the rotational direc-
84
tion are taken to be zero (∂/∂θ = 0). The governing equations, called the
85
axisymmetric Euler equations with swirl, are given by:
86
1 ∂rur ∂uz + = 0, r ∂r ∂z
(3)
1 ∂p ∂ur ∂ur u2θ ∂ur + ur − + uz = − , ∂t ∂r r ∂z ρ ∂r
(4)
∂uθ ur uθ ∂uθ ∂uθ + ur + + uz = 0, ∂t ∂r r ∂z
(5)
1 ∂p ∂uz ∂uz ∂uz + ur + uz = − , ∂t ∂r ∂z ρ ∂z
(6)
for continuity and
87
for (r), (θ) and (z)-momentum where ur , uθ and uz are the velocity compo-
88
nents in the radial, azimuthal and axial direction.
89
2.3. Boundary conditions and numerical procedure
90
The flow at the inlet (boundary S1 in figure 1) is modeled as a q-vortex.
91
The non-dimensional velocity profiles for the radial ur , azimuthal uθ and
92
axial uz velocity at the inlet are given by 6
Table 2: Mean relative difference between the 2D simulations and the reference grid of 180000 cells
δuz
δur
δuθ
δP
0.01%
0.05%
0.04%
0.04%
ur (r) = 0, and
Ωδ [1 r
uθ (r) =
− exp(−(r/δ)2 )]
(7)
uz (r) = 1
93
where Ω is the non-dimensional swirl ratio and δ = 1 is the core radius of
94
the vortex. Often, the amount of swirl is expressed by the swirl number Sw ,
95
which is the ratio of axial flux of azimuthal momentum and outer radius
96
times the axial flux of axial momentum,
Sw =
R Ri
uθ uz r2 dr . R Ri Ri 0 u2z rdr 0
(8)
97
Substitution of the velocity profiles given in equation 7 into equation 9 and
98
integrating between 0 and Ri results in
Sw =
Ωδ Ri2 + δ 2 exp(−(Ri /δ)2 ) − 1 , 3 Ri
(9)
99
which gives for the parameters in this study the relation Sw ≈ 0.3773Ω. At
100
the side and outlet boundaries (respectively S2 and S3 in figure 1), the gra-
101
dients off all variables in the normal direction of the boundary are taken to
102
be zero and boundary S4 is modeled as a symmetry axis. The numerical grid
103
consists of 71 cells in the radial and 625 cells in the axial direction, giving
104
a total number of 44 375 cells. The grid size is taken finer near the central 7
105
axis to resolve the gradients in the vortex core properly.
106
The equations of motion are solved using the finite volume method. A Mono-
107
tonic Upstream-Centered Scheme for Conservation Laws (MUSCL) scheme
108
is used for the spatial dicretisation as proposed by Van Leer (1979). For the
109
temporal discretisation, a second order implicit scheme is used. Although
110
this scheme is stable for all time steps chosen the maximum Courant number
111
in the domain is kept below 0.5. Iterations are stopped when the residuals
112
in each cell reach machine accuracy. The level of convergence (LCV), i.e.
113
the sum of the absolute values of the residuals in each cell, to reach machine
114
accuracy for each cell can be estimated as φ
115
Ntotal × 10−8 , (10) Ninlet where φ is the flux variable, Ni the number of inlet cells and Ntotal the total
116
number of celss. Given φ is the mass flow rate, this corresponds to a LCV
117
≈ 10−4 .
118
In order to assess the discretisation errors, the results are compared to the
119
solution on a 2D reference grid of around 180 000 cells. The results of the av-
120
erage relative error of a variable φ, defined as δφ = |φ2D − φref |/|φ2D |, where
121
the average is taken over all the grid points in the domain, is shown in table
122
2. The deviation with the reference grid is maximal 0.05% for both pressure
123
and velocities, indicating that the discretisation errors are sufficiently low.
124
2.4. Stability analysis
125
In case of axisymmetric steady flow, the momentum equations 4-6 can
126
be written as a single equation for the streamfunction Φ with the aid of the
127
vorticity equation as 8
∂ 2 Φ 1 ∂Φ ∂ 2 Φ dC 2 dH + − C , − = r ∂r2 r ∂r ∂z 2 dΦ dΦ
(11)
128
where the streamfunction Φ is given by uz = r−1 ∂Φ/∂r and ur = −r−1 ∂Φ/∂z.
129
This equation is called the Bragg-Hawthorne equation and links the stream-
130
function with the total head H = p/ρ + (u2r + u2θ + u2z )/2 and recirculation
131
C = ruθ . To check whether the flow can support wave perturbations, we de-
132
compose the streamfunction into a base flow Ψ and a very small perturbation
133
( << 1) as
Φ(r, z) = Ψ(r, z) + φ(r) exp(kz).
(12)
134
Substitution of this expression into equation 11, while linearising towards
135
results in 2
∂ φ − 1r ∂φ + ∂r ∂r2 ∂ (ru2 ) k 2 + r31u2 ∂r θ − z
136
This equation can be written as
r ∂ uz ∂r
1 ∂uz r ∂r
+
[C] · [Φ] = k 2 [Φ] ,
ur ∂ur ruz ∂r
φ = 0.
(13)
(14)
137
where the coefficient matrix [C] depends on the discretisation of the deriva-
138
tives in equation 13. In this study, both a second order central scheme is used
139
for the first and second order derivatives. The wavenumbers of the pertur-
140
bation function can be found by solving the eigenvalue problem in equation
141
14, where k 2 is the eigenvalue. If the eigenvalues are negative, the flow can
142
support standing waves. Note that Benjamin (1962) used a similar equation
9
(a) Axial velocity uz
(b) Azimuthal velocity uθ Figure 2: Velocity profiles of the 2D simulations for Ω=1.5.
143
to check whether a flow is subcritical or supercritical, where in his derivation
144
a columnar vortex is assumed, i.e. ur =0. In his theory on vortex breakdown,
145
a subcritical flow is able to sustain standing waves ( i.e. k 2 < 0), while a
146
supercritical flow is unable to support these waves (k 2 > 0) and breakdown is
147
then defined as the transition from a supercritical flow to a subcritical flow.
148
3. Results and discussion
149
Simulations with increasing Ω starting from 0 while keeping the profile
150
of uz constant (results not shown here) show that the transition from a flow
151
without vortex breakdown to a flow with breakdown is somewhere between
152
Ω = 1.5 and 1.52 or Sw = 0.566 and 0.573. This critical swirl number is
153
also found by Darmofal and Murman (1994) and Beran and Culick (1992) 10
Figure 3: Axial velocity uz at t∗ = 14/3 for Ω=1.52.
154
which performed viscous flow simulations with the same geometry and inlet
155
profiles for Reynolds numbers ranging from Re = 250 to 6000. For the
156
higher Reynolds numbers, the transitional swirl number’s variation is very
157
small, as confirmed by this study. For Ω = 1.5, the velocity fields of uz
158
and uθ are shown in figure 2. As the flow enters the convergent section of
159
the pipe (0 < z < 5), it is accelerated and the conservation of azimuthal
160
momentum increases uθ along the streamlines. In the divergent section of
161
the pipe (5 < z < 10), the flow is decelerated by a combination of the
162
positive axial pressure gradients induced by the divergent section of the pipe
163
and the decay of azimuthal velocity. In the constant section of the pipe
164
(10 < z < 25), the flow is quite uniform and finally in the convergent section
165
of the pipe (25 < z < 30) it is accelerated again towards the outlet. The
166
flowfield in figure 2 serves as the initial condition for the transient numerical
167
calculations were a sudden step increase from Ω = 1.5 to 1.52 is simulated.
168
In the following, time t is non-dimensionalised by the flowthrough time as
169
t∗ = tδuz (z = 0)/zmax .
170
After the sudden increase in inlet swirl, the step response of the flow has
171
died out after around t∗ = 14/3, i.e. about 5 flowthrough times. The axial
172
velocity field at that timestep is shown in figure 3. This flowfield is called 11
4
3
k 2 [rad2 /m2 ]
2
1
0 Increasing time -1
-2 0
5
10
15
20
25
30
z/δ [-]
Figure 4: The minimum eigenvalues |k 2 | at each axial location. The dashed line corre-
sponds to Ω = 1.5 and the first solid line corresponds to t∗ = 14/3. The timestep between two adjacent curves is ∆t∗ = 0.5, i.e. 0.5 flowthrough times.
173
pseudo-static, as the time derivatives of the variables in equations 4-6 are
174
very small compared to the other terms. For instance, considering the axial
175
momentum equation at the central axis, ∂uz ∂uz 1 ∂p = −uz − , ∂t ∂z ρ ∂z
(15)
176
shows that the maximal time derivative of the velocity, i.e. ∂uz /∂t is only
177
0.01% of the axial momentum-flux uz ∂uz /∂z. As the flow field at t∗ ≥ 14/3
178
is quasi-static, the time derivatives are negligible and solving the eigenvalue
179
problem in equation 13 for each axial location is a good approximation to
180
identify subcritical regions within the flow field and check whether stand-
181
ing waves can exist. The results are shown in figure 4, where the minimum
182
eigenvalue is plotted at each axial location. The dashed line corresponds to
183
Ω = 1.5. At this swirl number, the flow is supercritical in the entire domain
184
and no standing waves are supported. The solid lines corresponds to quasi-
185
static flow fields starting from t∗ = 14/3 in intervals of ∆t∗ = 0.5. The flow
186
at t∗ = 14/3 is subcritical between 11.2 < z/δ < 16.6 and supercritical in 12
187
the remainder of the domain. Figure 4 shows that the criterion of Benjamin
188
(1962) predicts very well the onset of vortex breakdown if applied locally as
189
also confirmed by other studies, even for viscous flow (Ruith et al., 2003).
190
Taking the quasi-static flow field at t∗ = 14/3 as a reference, the per-
191
turbation of the radial velocity u˜r (r, z, t∗ ) can be defined as u˜r (r, z, t∗ ) =
192
ur (r, z, t∗ ) − ur (r, z, t∗ = 14/3). The time evolution of this perturbation near
193
the central axis (radial location r/δ ≈ 0.007) is shown in figure 5. The profile
194
of u˜r corresponds to a wave which is located in the part of the domain where
195
the flow is subcritical (see figure 4). The perturbation grows exponentially
196
in time. The dimensionless growth rate of 0.83 is determined by exponential
197
fit of the maximum amplitude in figure 5b versus time. The wavelength of
198
u˜r corresponds to λ ≈ 2π/kmax , where kmax is the minimal wavenumber in
199
figure 4. Moreover, this wavelength is also more or less equal to the length
200
of the region where the flow is subcritical. This close relation between the
201
observed wavelengths and the critical region of the flow supports the theory
202
of Benjamin that vortex breakdown is the transition from supercritical to
203
subcritical flow.
204
As the perturbations in the flow grow near the central axis, an imbalance
205
is induced between the axial momentum in the flow and the static pressure
206
gradients in the flow direction near the central axis. Figure 5b shows this
207
imbalance as the left hand side of equation 15. The curves in the figure are
208
taken at the same time instants as the ones in figure 4. As the acceleration
209
of the flow is negative just upstream of the critical region, the flow is decel-
210
erated along the central axis. Due to this deceleration, the location where
211
the flow becomes subcritical moves upstream, as confirmed by figure 4. As
13
212
the imbalance grows exponentially in time, the deceleration near the central
213
axis increases, until a region of negative axial velocity is formed, called the
214
vortex breakdown bubble (figure 6). To the authors knowledge, this paper
215
would be the first one to identify this exponential growing imbalance in axial
216
momentum in a region where the flow is subcritical according to Benjamin
217
and this imbalance is the physical mechanism leading to the bubble forma-
218
tion. It is well known in literature that adverse pressure gradients decelerate
219
the flow near the central axis and promote the transition to breakdown, i.e.
220
they decrease the critical swirl number. However, looking at the velocity
221
fields in Figs. 2 and 3 shows that both flows are far from stagnant, i.e. the
222
axial velocity on the centerline is still in the order of 0.3-0.7 m/s. It is also
223
confirmed by Hall (1972) that flows near criticality are not necessary flows
224
near stagnation. Therefore, stagnation is a consequence of vortex breakdown
225
and vortex breakdown is not a consequence of stagnation, as also confirmed
226
by Cary and Darmofal (2001).
227
228
4. Conclusions
229
This paper studied the dynamics involved in the transition from a swirling
230
flow with swirl number below the onset of vortex breakdown to a swirling
231
flow undergoing breakdown. To eliminate the numerous parameters influ-
232
encing breakdown, the axisymmetric Euler equations with swirl are used as
233
a fluid flow model and solutions are obtained by means of numerical simu-
234
lation. It is shown that as the initial swirl increase has died out, the flow
235
evolves to a quasi-static state where time derivatives of variables are very 14
1
×10-3
0.015
∂uz /∂t[m/s2 ]
u ˜r (r ≈ 0.007, z, t∗ )[m/s]
λ/2 0.01
0.5
0
-0.5
Increasing time
-1
-1.5 0
0.005 0 -0.005
Increasing time
-0.01
5
10
15
20
25
-0.015
30
0
5
z/δ [-]
10
15
20
25
30
z/δ [-]
(a) Radial velocity perturbation at r/δ ≈ (b) Acceleration of the axial velocity at 0.007.
r/δ ≈ 0.007.
Figure 5: Perturbation of radial velocity and momentum imbalance near the central axis at r/δ ≈ 0.007. The first line corresponds to t∗ = 14/3. The timestep between two adjacent curves ∆t∗ = 0.5, i.e. 0.5 flowthrough times. The curves are taken at the same time instants as the solid curves in figure 4
Figure 6: Axial velocity fields at the transition to vortex breakdown. Axial velocity uz at t∗ = 10.
15
236
small. Stability analysis of this state shows that it can support standing
237
waves in a small region of the flow domain. As such, this study verifies the
238
criterion of Benjamin, which states that breakdown is the transition from a
239
supercritical flow to a subcritical flow. These standing waves are observed
240
in the simulations as an imbalance in the axial momentum equation which
241
slows down the flow near the central axis. The amplitude of this imbalance
242
grows exponentially in time, eventually leading to an axisymmetric recircu-
243
lation zone, called vortex breakdown. To the authors knowledge, this study
244
would be the first to reveal these features prior to breakdown and the results
245
may help in understanding of the physical mechanisms leading to breakdown
246
as this is still a controversial issue in literature.
247
5. References
248
Benjamin, T.B., 1962. Theory of the vortex breakdown phenomenon. J.
249
250
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Fluid Mech. 14, 593–629. Beran, P.S., Culick, F.E.C., 1992. The role of non-uniqueness in the development of vortex breakdown. J. Fluid Mech. 242, 491–527.
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Cary, A.W., Darmofal, D.L., 2001. Axisymmetric and Non-Axisymmetric
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Initiation of Vortex Breakdown. Technical Report. RTO AVT Symposium
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on ’Advanced Flow Management: Part A - Vortex Flows and High Angle
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Darmofal, D.L., Murman, E.M., 1994. On the trapped wave nature of axisymmetric vortex breakdown. AIAA Paper 94-2318 .
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Escudier, M.P., 1988. Vortex breakdown: observations and explanations. Prog. Aerospace Sci. 25, 189–229. Gallaire, F., Chomaz, J.M., 2004. The role of boundary conditions in a simple model of incipient vortex breakdown. Phys. Fluids 16, 274–286. Gallaire, F., Ruith, M., Meiburg, E., Chomaz, J.M., Huerre, P., 2006. Spiral vortex breakdown as a global mode. J. Fluid Mech. 549, 71–80. Hall, M.G., 1972. Vortex breakdown. Annual Review of Fluid Mechanics 4, 195–217.
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some observations and discussion of the phenomenon. Technical Report
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Liang, H., Maxworthy, T., 2005. An experimental investigation of swirling jets. J. Fluid Mech. 525, 115–159. Lucca-Negro, O., O’Doherty, T., 2001. Vortex breakdown: a review. Prog. Energy Combust. Sci. 27, 431–481. Ludwieg, H., 1960. Stabilitat der stromung in einem zylindrischen ringraum. Z. Flugwiss. 8, 135–140.
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Meliga, P., Gallaire, F., Chomaz, J.M., 2012. A weakly nonlinear mechanism for mode selection in swirling jets. J. Fluid Mech. 699, 216–262. 17
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Noack, B., Wygnanski, I., 2011. Three-dimensional coherent structures in a
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Peckham, D.H., Atkinson, S.A., 1957. Preliminary results of low speed wind
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tunnel test on a Ghotic wing of aspect ration 1.0. Technical Report. British
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Qadri, U., Mistry, D., Juniper, M., 2013. Structural sensitivity of spiral vortex breakdown. J. Fluid Mech. 720, 558–581.
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Ruith, M., Chen, P., Meiburg, E., Maxworthy, T., 2003. Three-dimensional
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vortex breakdown in swirling jets and wakes. J. Fluid Mech. 486, 331–378.
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Van Leer, B., 1979. Toward the ultimate concervative difference scheme. iv. a
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second order sequel to godunov’s method. J. Comput. Phys. 32, 101–136.
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Wang, S., Rusak, Z., 1996. On the stability of an axisymmetric rotating flow
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in a pipe. Phys. Fluids 8, 1007–1016.
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Wang, S., Rusak, Z., 1997. The dynamics of a swirling flow in a pipe and
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transition to axisymmetric vortex breakdown. J. Fluid Mech. 340, 177–223.
18
S0997-7546(16)30101-7 http://dx.doi.org/10.1016/j.euromechflu.2017.02.009 EJMFLU 3144
To appear in:
European Journal of Mechanics B/Fluids
Received date: 22 March 2016 Please cite this article as: M. Vanierschot, On the dynamics of the transition to vortex breakdown in axisymmetric inviscid swirling flows, European Journal of Mechanics B/Fluids (2017), http://dx.doi.org/10.1016/j.euromechflu.2017.02.009 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
2
On the dynamics of the transition to vortex breakdown in axisymmetric inviscid swirling flows
3
M. Vanierschota
1
4 5
6
a
KU Leuven, campus Group T, Andreas Vesaliusstraat 13, B-3000 Leuven, Belgium. Email: [email protected]
Abstract This paper reports on novel features found in the dynamics of the transition to vortex breakdown in inviscid axisymmetric flows with swirl. These features are revealed by a transient simulation of an open ended pipe flow where the inlet swirl is suddenly increased from a swirl number just below the onset of vortex breakdown to a swirl number just above the onset of vortex breakdown. To eliminate the numerous parameters influencing breakdown, the axisymmetric Euler equations with swirl are used as a fluid flow model and solutions are obtained by means of numerical simulation. It is shown that as the step response has died out, the flow evolves to a quasi-static state where time derivatives of variables are negligible small. Stability analysis of this state shows that it can support standing waves in a small region of the flow domain. These standing waves are observed in the simulations as an imbalance in the axial momentum equation which slows down the flow near the central axis. The amplitude of this imbalance grows exponentially in time with a dimensionless growth rate of 0.83 scaled with the flowthrough time. Eventually, the axial velocity along the central axis becomes negative in a small part of the flow, leading to an axisymmetric recirculation zone,
Preprint submitted to International Journal of Heat and Fluid Flow
November 11, 2016
called vortex breakdown. To the authors knowledge, this study would be the first to reveal these features prior to breakdown and the results may help in understanding of the physical mechanisms leading to it as this is still a controversial issue in literature. 7
Keywords: Vortex breakdown, Stability analysis, Inviscid swirling flow
8
1. Introduction
9
Vortex breakdown has fascinated the scientific community for almost six
10
decades now. It has firstly been discovered by Peckham and Atkinson (1957)
11
as the bursting of a leading-edge vortex in the flow over a delta wing. Later it
12
has also been found in numerous other engineering flow cases, amongst others
13
rotating pipe flow, jet flow and enclosed cylinder flow. Despite research for
14
over six decades now, there is still no general explanation for vortex break-
15
down as it is a complex function of numerous flow parameters. Two main
16
general theories exist: the instability theory of Ludwieg (1960) and the tran-
17
sition from a supercritical to a subcritical flow by Benjamin (1962). Ludwieg
18
stated that vortex breakdown is a consequence of hydrodynamic instabilities
19
in the flow which grow in time and eventually lead to breakdown (Ludwieg,
20
1960). Benjamin stated that breakdown is the transition of a supercritical
21
flow (unable to support standing waves) to a subcritical flow (able to support
22
standing waves), similar to the hydraulic jump (Benjamin, 1962). As both
23
theories are unable to predict all the features of breakdown found in experi-
24
ments and simulations, both have found no general acceptance in literature
25
as the main mechanism leading to breakdown.
26
The main reason for the lack of a general theory is the fact that vortex
2
27
breakdown manifests itself in many forms which are depend upon many pa-
28
rameters. In experimental work, no less than 7 types have been identified.
29
The two most commonly observed, called bubble and spiral breakdown, were
30
first reported by Lambourne and Bryer (1961). Bubble or spiral breakdown
31
have been observed in different experiments with similar settings and even
32
transitions between them within the same experiment (without changing the
33
inflow parameters) have been reported. This has led to disagreement on the
34
origin of breakdown (see for instance the review papers of Escudier (1988)
35
and Lucca-Negro and O’Doherty (2001)). Recent studies showed some more
36
insight in the mechanism leading to breakdown. It was found that flows
37
going from below to above the critical state become unstable (Gallaire and
38
Chomaz, 2004; Wang and Rusak, 1996, 1997). Moreover, it has been shown
39
by numerous authors that spiral breakdown occurs in the wake of an ax-
40
isymmetric breakdown as a global instability mode of the flow (Liang and
41
Maxworthy, 2005; Ruith et al., 2003; Gallaire et al., 2006; Meliga et al., 2012;
42
Qadri et al., 2013; Luginsland et al., 2016; Oberleithner et al., 2011). Two
43
modes of spiral breakdown have been observed: the single helix (|m|=1) and
44
the double helix (|m|=2), where m is the azimuthal wave number. A re-
45
cent study of Meliga et al. (2012) showed that both single or double helix
46
breakdown are a bifurcation from axisymmetric breakdown and that mode
47
selection depends on the swirl number.
48
In helping to understand the physical phenomena leading to breakdown, this
49
paper studies the dynamics of an axisymmetric inviscid flow just before the
50
onset of vortex breakdown. Analysis of the axial and radial momentum bal-
51
ances and a stability analysis reveals the mechanisms leading to the forma-
3
52
tion of an axisymmetric breakdown bubble. As such, the results of this study
53
may contribute to more understanding of the physical mechanisms leading
54
to vortex breakdown.
55
2. Governing equations and boundary conditions
56
2.1. Pipe geometry
57
The geometry used in this study is based on the numerical study of Dar-
58
mofal and Murman (1994) and the experimental work of Sarpkaya (1974)
59
and is also similar to the one used in Beran and Culick (1992). A schematic
60
view is shown in figure 1. The geometry is rotational symmetric in the θ di-
61
rection in a cylindrical (r,θ,z)-coordinate system, where r is the radial, θ the
62
azimuthal and z the axial direction. The pipe is divided into four different
63
sections and the radius of the pipe R(z) as a function of the axial direction
64
z is given as
R(z) =
Ri + (Rt − Ri )g(z, zt ), Rt + (Ro − Rt )g(z − zt , zo − zt ),
(0 < z < zt ) (zt < z < zo )
Ro , (zo < z < zc ) R + (R c max − Ro )g(z − zc , zmax − zc ), (zc < z < zmax )
(1)
65
where the function g(z, L) is defined as
66
1 g(z, L) = {1 − cos[π(z/L)]} (2) 2 and the values of the dimensions are given in table 1. The first section
67
(0 < z < zt ) is convergent which ensures a supercritical flow at zt to pre-
68
vent disturbances from traveling upstream towards the inlet section, i.e. it 4
12
10
8
r
S2
6
4
Section II
Section I
S1
Section IV
Section III
S3
2
0
zo
zt
zc
S4
zmax
z
−2
−4
−6
Figure 1: Schematic view of the pipe geometry and the different sections. The radial direction is enlarged by a factor 3 for clarity. The radii in Table 1 corresponding to the axial locations z in the figure have the same index. 0
5
10
15
20
25
30
Table 1: Values of the pipe geometry constants
zi
zt
zo
zc
zmax
0
5
10
25
30
Ri
Rt
Ro
Rc
Rmax
2.0 1.8
2.0
2.0
1.8
69
separates the inlet from the vortex breakdown region. The second section
70
(zt < z < zo ) is diverging to create an adverse pressure gradient on the flow.
71
Adverse pressure gradients are known to promote vortex breakdown (Sarp-
72
kaya, 1974) and hence the location of the bubble lies within or just after this
73
section. The third section (zo < z < zc ) has a constant radius and the last
74
section (zc < z < zmax ) is convergent. This promotes a transition of the
75
flow towards the supercritical state again and hence preventing the output
76
boundary conditions from disturbing the bubble formation.
5
35
40
77
2.2. Governing equations
78
Despite the fact that most swirling flows are highly three-dimensional,
79
breakdown often starts axisymmetric, especially for low Reynolds number
80
flows. After the onset of axisymmetric breakdown, helical instabilities start
81
to develop, creating a three-dimensional flow field (e.g. Ruith et al. (2003)).
82
As this study focuses on the features before breakdown, the axisymmetric
83
variant of the equations are taken were derivatives in the rotational direc-
84
tion are taken to be zero (∂/∂θ = 0). The governing equations, called the
85
axisymmetric Euler equations with swirl, are given by:
86
1 ∂rur ∂uz + = 0, r ∂r ∂z
(3)
1 ∂p ∂ur ∂ur u2θ ∂ur + ur − + uz = − , ∂t ∂r r ∂z ρ ∂r
(4)
∂uθ ur uθ ∂uθ ∂uθ + ur + + uz = 0, ∂t ∂r r ∂z
(5)
1 ∂p ∂uz ∂uz ∂uz + ur + uz = − , ∂t ∂r ∂z ρ ∂z
(6)
for continuity and
87
for (r), (θ) and (z)-momentum where ur , uθ and uz are the velocity compo-
88
nents in the radial, azimuthal and axial direction.
89
2.3. Boundary conditions and numerical procedure
90
The flow at the inlet (boundary S1 in figure 1) is modeled as a q-vortex.
91
The non-dimensional velocity profiles for the radial ur , azimuthal uθ and
92
axial uz velocity at the inlet are given by 6
Table 2: Mean relative difference between the 2D simulations and the reference grid of 180000 cells
δuz
δur
δuθ
δP
0.01%
0.05%
0.04%
0.04%
ur (r) = 0, and
Ωδ [1 r
uθ (r) =
− exp(−(r/δ)2 )]
(7)
uz (r) = 1
93
where Ω is the non-dimensional swirl ratio and δ = 1 is the core radius of
94
the vortex. Often, the amount of swirl is expressed by the swirl number Sw ,
95
which is the ratio of axial flux of azimuthal momentum and outer radius
96
times the axial flux of axial momentum,
Sw =
R Ri
uθ uz r2 dr . R Ri Ri 0 u2z rdr 0
(8)
97
Substitution of the velocity profiles given in equation 7 into equation 9 and
98
integrating between 0 and Ri results in
Sw =
Ωδ Ri2 + δ 2 exp(−(Ri /δ)2 ) − 1 , 3 Ri
(9)
99
which gives for the parameters in this study the relation Sw ≈ 0.3773Ω. At
100
the side and outlet boundaries (respectively S2 and S3 in figure 1), the gra-
101
dients off all variables in the normal direction of the boundary are taken to
102
be zero and boundary S4 is modeled as a symmetry axis. The numerical grid
103
consists of 71 cells in the radial and 625 cells in the axial direction, giving
104
a total number of 44 375 cells. The grid size is taken finer near the central 7
105
axis to resolve the gradients in the vortex core properly.
106
The equations of motion are solved using the finite volume method. A Mono-
107
tonic Upstream-Centered Scheme for Conservation Laws (MUSCL) scheme
108
is used for the spatial dicretisation as proposed by Van Leer (1979). For the
109
temporal discretisation, a second order implicit scheme is used. Although
110
this scheme is stable for all time steps chosen the maximum Courant number
111
in the domain is kept below 0.5. Iterations are stopped when the residuals
112
in each cell reach machine accuracy. The level of convergence (LCV), i.e.
113
the sum of the absolute values of the residuals in each cell, to reach machine
114
accuracy for each cell can be estimated as φ
115
Ntotal × 10−8 , (10) Ninlet where φ is the flux variable, Ni the number of inlet cells and Ntotal the total
116
number of celss. Given φ is the mass flow rate, this corresponds to a LCV
117
≈ 10−4 .
118
In order to assess the discretisation errors, the results are compared to the
119
solution on a 2D reference grid of around 180 000 cells. The results of the av-
120
erage relative error of a variable φ, defined as δφ = |φ2D − φref |/|φ2D |, where
121
the average is taken over all the grid points in the domain, is shown in table
122
2. The deviation with the reference grid is maximal 0.05% for both pressure
123
and velocities, indicating that the discretisation errors are sufficiently low.
124
2.4. Stability analysis
125
In case of axisymmetric steady flow, the momentum equations 4-6 can
126
be written as a single equation for the streamfunction Φ with the aid of the
127
vorticity equation as 8
∂ 2 Φ 1 ∂Φ ∂ 2 Φ dC 2 dH + − C , − = r ∂r2 r ∂r ∂z 2 dΦ dΦ
(11)
128
where the streamfunction Φ is given by uz = r−1 ∂Φ/∂r and ur = −r−1 ∂Φ/∂z.
129
This equation is called the Bragg-Hawthorne equation and links the stream-
130
function with the total head H = p/ρ + (u2r + u2θ + u2z )/2 and recirculation
131
C = ruθ . To check whether the flow can support wave perturbations, we de-
132
compose the streamfunction into a base flow Ψ and a very small perturbation
133
( << 1) as
Φ(r, z) = Ψ(r, z) + φ(r) exp(kz).
(12)
134
Substitution of this expression into equation 11, while linearising towards
135
results in 2
∂ φ − 1r ∂φ + ∂r ∂r2 ∂ (ru2 ) k 2 + r31u2 ∂r θ − z
136
This equation can be written as
r ∂ uz ∂r
1 ∂uz r ∂r
+
[C] · [Φ] = k 2 [Φ] ,
ur ∂ur ruz ∂r
φ = 0.
(13)
(14)
137
where the coefficient matrix [C] depends on the discretisation of the deriva-
138
tives in equation 13. In this study, both a second order central scheme is used
139
for the first and second order derivatives. The wavenumbers of the pertur-
140
bation function can be found by solving the eigenvalue problem in equation
141
14, where k 2 is the eigenvalue. If the eigenvalues are negative, the flow can
142
support standing waves. Note that Benjamin (1962) used a similar equation
9
(a) Axial velocity uz
(b) Azimuthal velocity uθ Figure 2: Velocity profiles of the 2D simulations for Ω=1.5.
143
to check whether a flow is subcritical or supercritical, where in his derivation
144
a columnar vortex is assumed, i.e. ur =0. In his theory on vortex breakdown,
145
a subcritical flow is able to sustain standing waves ( i.e. k 2 < 0), while a
146
supercritical flow is unable to support these waves (k 2 > 0) and breakdown is
147
then defined as the transition from a supercritical flow to a subcritical flow.
148
3. Results and discussion
149
Simulations with increasing Ω starting from 0 while keeping the profile
150
of uz constant (results not shown here) show that the transition from a flow
151
without vortex breakdown to a flow with breakdown is somewhere between
152
Ω = 1.5 and 1.52 or Sw = 0.566 and 0.573. This critical swirl number is
153
also found by Darmofal and Murman (1994) and Beran and Culick (1992) 10
Figure 3: Axial velocity uz at t∗ = 14/3 for Ω=1.52.
154
which performed viscous flow simulations with the same geometry and inlet
155
profiles for Reynolds numbers ranging from Re = 250 to 6000. For the
156
higher Reynolds numbers, the transitional swirl number’s variation is very
157
small, as confirmed by this study. For Ω = 1.5, the velocity fields of uz
158
and uθ are shown in figure 2. As the flow enters the convergent section of
159
the pipe (0 < z < 5), it is accelerated and the conservation of azimuthal
160
momentum increases uθ along the streamlines. In the divergent section of
161
the pipe (5 < z < 10), the flow is decelerated by a combination of the
162
positive axial pressure gradients induced by the divergent section of the pipe
163
and the decay of azimuthal velocity. In the constant section of the pipe
164
(10 < z < 25), the flow is quite uniform and finally in the convergent section
165
of the pipe (25 < z < 30) it is accelerated again towards the outlet. The
166
flowfield in figure 2 serves as the initial condition for the transient numerical
167
calculations were a sudden step increase from Ω = 1.5 to 1.52 is simulated.
168
In the following, time t is non-dimensionalised by the flowthrough time as
169
t∗ = tδuz (z = 0)/zmax .
170
After the sudden increase in inlet swirl, the step response of the flow has
171
died out after around t∗ = 14/3, i.e. about 5 flowthrough times. The axial
172
velocity field at that timestep is shown in figure 3. This flowfield is called 11
4
3
k 2 [rad2 /m2 ]
2
1
0 Increasing time -1
-2 0
5
10
15
20
25
30
z/δ [-]
Figure 4: The minimum eigenvalues |k 2 | at each axial location. The dashed line corre-
sponds to Ω = 1.5 and the first solid line corresponds to t∗ = 14/3. The timestep between two adjacent curves is ∆t∗ = 0.5, i.e. 0.5 flowthrough times.
173
pseudo-static, as the time derivatives of the variables in equations 4-6 are
174
very small compared to the other terms. For instance, considering the axial
175
momentum equation at the central axis, ∂uz ∂uz 1 ∂p = −uz − , ∂t ∂z ρ ∂z
(15)
176
shows that the maximal time derivative of the velocity, i.e. ∂uz /∂t is only
177
0.01% of the axial momentum-flux uz ∂uz /∂z. As the flow field at t∗ ≥ 14/3
178
is quasi-static, the time derivatives are negligible and solving the eigenvalue
179
problem in equation 13 for each axial location is a good approximation to
180
identify subcritical regions within the flow field and check whether stand-
181
ing waves can exist. The results are shown in figure 4, where the minimum
182
eigenvalue is plotted at each axial location. The dashed line corresponds to
183
Ω = 1.5. At this swirl number, the flow is supercritical in the entire domain
184
and no standing waves are supported. The solid lines corresponds to quasi-
185
static flow fields starting from t∗ = 14/3 in intervals of ∆t∗ = 0.5. The flow
186
at t∗ = 14/3 is subcritical between 11.2 < z/δ < 16.6 and supercritical in 12
187
the remainder of the domain. Figure 4 shows that the criterion of Benjamin
188
(1962) predicts very well the onset of vortex breakdown if applied locally as
189
also confirmed by other studies, even for viscous flow (Ruith et al., 2003).
190
Taking the quasi-static flow field at t∗ = 14/3 as a reference, the per-
191
turbation of the radial velocity u˜r (r, z, t∗ ) can be defined as u˜r (r, z, t∗ ) =
192
ur (r, z, t∗ ) − ur (r, z, t∗ = 14/3). The time evolution of this perturbation near
193
the central axis (radial location r/δ ≈ 0.007) is shown in figure 5. The profile
194
of u˜r corresponds to a wave which is located in the part of the domain where
195
the flow is subcritical (see figure 4). The perturbation grows exponentially
196
in time. The dimensionless growth rate of 0.83 is determined by exponential
197
fit of the maximum amplitude in figure 5b versus time. The wavelength of
198
u˜r corresponds to λ ≈ 2π/kmax , where kmax is the minimal wavenumber in
199
figure 4. Moreover, this wavelength is also more or less equal to the length
200
of the region where the flow is subcritical. This close relation between the
201
observed wavelengths and the critical region of the flow supports the theory
202
of Benjamin that vortex breakdown is the transition from supercritical to
203
subcritical flow.
204
As the perturbations in the flow grow near the central axis, an imbalance
205
is induced between the axial momentum in the flow and the static pressure
206
gradients in the flow direction near the central axis. Figure 5b shows this
207
imbalance as the left hand side of equation 15. The curves in the figure are
208
taken at the same time instants as the ones in figure 4. As the acceleration
209
of the flow is negative just upstream of the critical region, the flow is decel-
210
erated along the central axis. Due to this deceleration, the location where
211
the flow becomes subcritical moves upstream, as confirmed by figure 4. As
13
212
the imbalance grows exponentially in time, the deceleration near the central
213
axis increases, until a region of negative axial velocity is formed, called the
214
vortex breakdown bubble (figure 6). To the authors knowledge, this paper
215
would be the first one to identify this exponential growing imbalance in axial
216
momentum in a region where the flow is subcritical according to Benjamin
217
and this imbalance is the physical mechanism leading to the bubble forma-
218
tion. It is well known in literature that adverse pressure gradients decelerate
219
the flow near the central axis and promote the transition to breakdown, i.e.
220
they decrease the critical swirl number. However, looking at the velocity
221
fields in Figs. 2 and 3 shows that both flows are far from stagnant, i.e. the
222
axial velocity on the centerline is still in the order of 0.3-0.7 m/s. It is also
223
confirmed by Hall (1972) that flows near criticality are not necessary flows
224
near stagnation. Therefore, stagnation is a consequence of vortex breakdown
225
and vortex breakdown is not a consequence of stagnation, as also confirmed
226
by Cary and Darmofal (2001).
227
228
4. Conclusions
229
This paper studied the dynamics involved in the transition from a swirling
230
flow with swirl number below the onset of vortex breakdown to a swirling
231
flow undergoing breakdown. To eliminate the numerous parameters influ-
232
encing breakdown, the axisymmetric Euler equations with swirl are used as
233
a fluid flow model and solutions are obtained by means of numerical simu-
234
lation. It is shown that as the initial swirl increase has died out, the flow
235
evolves to a quasi-static state where time derivatives of variables are very 14
1
×10-3
0.015
∂uz /∂t[m/s2 ]
u ˜r (r ≈ 0.007, z, t∗ )[m/s]
λ/2 0.01
0.5
0
-0.5
Increasing time
-1
-1.5 0
0.005 0 -0.005
Increasing time
-0.01
5
10
15
20
25
-0.015
30
0
5
z/δ [-]
10
15
20
25
30
z/δ [-]
(a) Radial velocity perturbation at r/δ ≈ (b) Acceleration of the axial velocity at 0.007.
r/δ ≈ 0.007.
Figure 5: Perturbation of radial velocity and momentum imbalance near the central axis at r/δ ≈ 0.007. The first line corresponds to t∗ = 14/3. The timestep between two adjacent curves ∆t∗ = 0.5, i.e. 0.5 flowthrough times. The curves are taken at the same time instants as the solid curves in figure 4
Figure 6: Axial velocity fields at the transition to vortex breakdown. Axial velocity uz at t∗ = 10.
15
236
small. Stability analysis of this state shows that it can support standing
237
waves in a small region of the flow domain. As such, this study verifies the
238
criterion of Benjamin, which states that breakdown is the transition from a
239
supercritical flow to a subcritical flow. These standing waves are observed
240
in the simulations as an imbalance in the axial momentum equation which
241
slows down the flow near the central axis. The amplitude of this imbalance
242
grows exponentially in time, eventually leading to an axisymmetric recircu-
243
lation zone, called vortex breakdown. To the authors knowledge, this study
244
would be the first to reveal these features prior to breakdown and the results
245
may help in understanding of the physical mechanisms leading to breakdown
246
as this is still a controversial issue in literature.
247
5. References
248
Benjamin, T.B., 1962. Theory of the vortex breakdown phenomenon. J.
249
250
251
Fluid Mech. 14, 593–629. Beran, P.S., Culick, F.E.C., 1992. The role of non-uniqueness in the development of vortex breakdown. J. Fluid Mech. 242, 491–527.
252
Cary, A.W., Darmofal, D.L., 2001. Axisymmetric and Non-Axisymmetric
253
Initiation of Vortex Breakdown. Technical Report. RTO AVT Symposium
254
on ’Advanced Flow Management: Part A - Vortex Flows and High Angle
255
of Attack for Military Vehicles’.
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257
Darmofal, D.L., Murman, E.M., 1994. On the trapped wave nature of axisymmetric vortex breakdown. AIAA Paper 94-2318 .
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Escudier, M.P., 1988. Vortex breakdown: observations and explanations. Prog. Aerospace Sci. 25, 189–229. Gallaire, F., Chomaz, J.M., 2004. The role of boundary conditions in a simple model of incipient vortex breakdown. Phys. Fluids 16, 274–286. Gallaire, F., Ruith, M., Meiburg, E., Chomaz, J.M., Huerre, P., 2006. Spiral vortex breakdown as a global mode. J. Fluid Mech. 549, 71–80. Hall, M.G., 1972. Vortex breakdown. Annual Review of Fluid Mechanics 4, 195–217.
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Lambourne, N.C., Bryer, D.W., 1961. The bursting of leading-edge vortices:
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some observations and discussion of the phenomenon. Technical Report
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Liang, H., Maxworthy, T., 2005. An experimental investigation of swirling jets. J. Fluid Mech. 525, 115–159. Lucca-Negro, O., O’Doherty, T., 2001. Vortex breakdown: a review. Prog. Energy Combust. Sci. 27, 431–481. Ludwieg, H., 1960. Stabilitat der stromung in einem zylindrischen ringraum. Z. Flugwiss. 8, 135–140.
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Meliga, P., Gallaire, F., Chomaz, J.M., 2012. A weakly nonlinear mechanism for mode selection in swirling jets. J. Fluid Mech. 699, 216–262. 17
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Noack, B., Wygnanski, I., 2011. Three-dimensional coherent structures in a
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Peckham, D.H., Atkinson, S.A., 1957. Preliminary results of low speed wind
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tunnel test on a Ghotic wing of aspect ration 1.0. Technical Report. British
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Qadri, U., Mistry, D., Juniper, M., 2013. Structural sensitivity of spiral vortex breakdown. J. Fluid Mech. 720, 558–581.
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